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%I #15 Feb 14 2024 20:03:37
%S 1,-1,-1,1,2,1,-1,-3,-3,-1,0,3,6,4,1,1,0,-7,-10,-5,-1,-1,-6,-1,14,15,
%T 6,1,0,12,25,3,-25,-21,-7,-1,0,-14,-64,-75,-5,41,28,8,1,1,11,102,231,
%U 179,5,-63,-36,-9,-1,-1,-4,-109,-448,-651,-365,0,92,45,10,1,0,-5,53,593,1486,1546,665,-14,-129,-55,-11,-1,0,12,75,-407,-2342,-4077,-3241,-1114,42,175,66,12,1
%N Expansion of g.f. A(x,y) satisfying Sum_{n>=0} Product_{k=1..n} (x^k + y*A(x,y)) = 1 + (y+1) * Sum_{n>=1} x^(n*(n+1)/2), as a triangle of coefficients T(n,k) of x^n*y^k in A(x,y) read by rows.
%C A370141(n) = Sum_{k=0..n-1} T(n,k), for n >= 1.
%C A370142(n) = Sum_{k=0..n-1} T(n,k) * 2^k, for n >= 1.
%C A370143(n) = Sum_{k=0..n-1} T(n,k) * 3^k, for n >= 1.
%C A370144(n) = Sum_{k=0..n-1} T(n,k) * 4^k, for n >= 1.
%H Paul D. Hanna, <a href="/A370140/b370140.txt">Table of n, a(n) for n = 1..2485</a>
%F G.f. A(x,y) = Sum_{n>=1} Sum_{k=0..n-1} T(n,k)*x^n*y^k satisfies the following formulas.
%F Let Q(x,y) = 1 + (y+1) * Sum_{n>=1} x^(n*(n+1)/2), then
%F (1) Q(x,y) = Sum_{n>=0} Product_{k=1..n} (x^k + y*A(x,y)).
%F (2) Q(x,y) = Sum_{n>=0} x^(n*(n+1)/2) / Product_{k=0..n} (1 - x^k * y*A(x,y)).
%F (3) Q(x,y) = 1/(1 - F(1)), where F(n) = (x^n + y*A(x,y))/(1 + x^n + y*A(x,y) - F(n+1)), a continued fraction.
%F (4) A(x,0) = (1-x) * Sum_{n>=1} x^(n*(n+1)/2), which is the g.f. of column 0.
%F (5) d/dy A(x,y) at y = 0 equals -(1-x)^2/(1+x) * Q(x,0) * (Q(x,0) - 1)^2, which is the g.f. of column 1.
%e G.f.: A(x,y) = x*(1) + x^2*(-1 - y) + x^3*(1 + 2*y + y^2) + x^4*(-1 - 3*y - 3*y^2 - y^3) + x^5*(3*y + 6*y^2 + 4*y^3 + y^4) + x^6*(1 - 7*y^2 - 10*y^3 - 5*y^4 - y^5) + x^7*(-1 - 6*y - y^2 + 14*y^3 + 15*y^4 + 6*y^5 + y^6) + x^8*(12*y + 25*y^2 + 3*y^3 - 25*y^4 - 21*y^5 - 7*y^6 - y^7) + x^9*(-14*y - 64*y^2 - 75*y^3 - 5*y^4 + 41*y^5 + 28*y^6 + 8*y^7 + y^8) + x^10*(1 + 11*y + 102*y^2 + 231*y^3 + 179*y^4 + 5*y^5 - 63*y^6 - 36*y^7 - 9*y^8 - y^9) + x^11*(-1 - 4*y - 109*y^2 - 448*y^3 - 651*y^4 - 365*y^5 + 92*y^7 + 45*y^8 + 10*y^9 + y^10) + ...
%e Let Q(x,y) = 1 + (y+1) * Sum_{n>=1} x^(n*(n+1)/2)
%e then A = A(x,y) satisfies
%e (1) Q(x,y) = 1 + (x + y*A) + (x + y*A)*(x^2 + y*A) + (x + y*A)*(x^2 + y*A)*(x^3 + y*A) + (x + y*A)*(x^2 + y*A)*(x^3 + y*A)*(x^4 + y*A) + (x + y*A)*(x^2 + y*A)*(x^3 + y*A)*(x^4 + y*A)*(x^5 + y*A) + ...
%e also
%e (2) Q(x,y) = 1/(1 - y*A) + x/((1 - y*A)*(1 - x*y*A)) + x^3/((1 - y*A)*(1 - x*y*A)*(1 - x^2*y*A)) + x^6/((1 - y*A)*(1 - x*y*A)*(1 - x^2*y*A)*(1 - x^3*y*A)) + x^10/((1 - y*A)*(1 - x*y*A)*(1 - x^2*y*A)*(1 - x^3*y*A)*(1 - x^4*y*A)) + ...
%e Further, A = A(x) satisfies the continued fraction given by
%e (3) Q(x,y) = 1/(1 - (x + y*A)/(1 + x + y*A - (x^2 + y*A)/(1 + x^2 + y*A - (x^3 + y*A)/(1 + x^3 + y*A - (x^4 + y*A)/(1 + x^4 + y*A - (x^5 + y*A)/(1 + x^5 + y*A - (x^6 + y*A)/(1 + x^6 + y*A - (x^7 + y*A)/(1 - ...))))))))
%e where
%e Q(x,y) = 1 + (y+1)*x + (y+1)*x^3 + (y+1)*x^6 + (y+1)*x^10 + (y+1)*x^15 + (y+1)*x^21 + ... + (y+1)*x^(n*(n+1)/2) + ...
%e This triangle of coefficients T(n,k) of x^n*y^k in A(x,y) begins
%e 1;
%e -1, -1;
%e 1, 2, 1;
%e -1, -3, -3, -1;
%e 0, 3, 6, 4, 1;
%e 1, 0, -7, -10, -5, -1;
%e -1, -6, -1, 14, 15, 6, 1;
%e 0, 12, 25, 3, -25, -21, -7, -1;
%e 0, -14, -64, -75, -5, 41, 28, 8, 1;
%e 1, 11, 102, 231, 179, 5, -63, -36, -9, -1;
%e -1, -4, -109, -448, -651, -365, 0, 92, 45, 10, 1;
%e 0, -5, 53, 593, 1486, 1546, 665, -14, -129, -55, -11, -1;
%e 0, 12, 75, -407, -2342, -4077, -3241, -1114, 42, 175, 66, 12, 1;
%e 0, -15, -240, -391, 2087, 7481, 9736, 6182, 1749, -90, -231, -78, -13, -1;
%e 1, 19, 375, 1867, 1232, -8239, -20469, -20908, -10953, -2608, 165, 298, 91, 14, 1;
%e -1, -26, -420, -3629, -9480, -2256, 26993, 49721, 41292, 18292, 3729, -275, -377, -105, -15, -1;
%e ...
%e The g.f. of column 0 = (1-x) * Sum_{n>=1} x^(n*(n+1)/2).
%e The g.f. of column 1 = -(1-x)^2/(1+x) * [Sum_{n>=0} x^(n*(n+1)/2)] * [Sum_{n>=1} x^(n*(n+1)/2)]^2.
%o (PARI) {T(n,k) = my(A=[0,1]); for(i=1,n, A = concat(A,0);
%o A[#A] = polcoeff( (sum(m=1,#A, prod(k=1,m, x^k + y*Ser(A) ) ) - (y+1)*sum(m=1,sqrtint(2*#A+1), x^(m*(m+1)/2) ) )/(-y), #A-1) ); H=A; polcoeff(A[n+1],k,y)}
%o for(n=1,21, for(k=0,n-1, print1(T(n,k),", "));print(""))
%o (PARI) /* Generate A(x,y) recursively using integration wrt y */
%o {T(n,k) = my(A = x +x*O(x^n), Q = sum(m=1,sqrtint(2*n+1), x^(m*(m+1)/2) +x*O(x^n)) );
%o for(i=1,n, A = (1/y) * intformal( Q/sum(m=1,n, sum(j=1,m, prod(k=1,m, if(j==k,1, x^k + y*A) +O(x^n))) ), y) ); polcoeff(polcoeff(A,n,x),k,y)}
%o for(n=1,15,for(k=0,n-1, print1(T(n,k),", "));print(""))
%Y Cf. A370141 (y=1), A370142 (y=2), A370143 (y=3), A370144 (y=4).
%Y Cf. A370145 (column 1), A370146 (column 2).
%K sign,tabl
%O 1,5
%A _Paul D. Hanna_, Feb 12 2024
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